- #1

- 10

- 0

## Homework Statement

Prove that limx→2 (1/(x – 2)) does not exist.

**2. The attempt at a solution**

My approach to this problem is to use the sequential criterion for nonexistence, that is, if {x_n} and {y_n} both converge to a (2 in this case), but limx→a f(x_n) is not equal to limx→a f(y_n), then limx→a f(x) does not exist (a being 2 in this case).

I was able to find two sequences, {x_n} and {y_n}, both converging to 2 (I chose {x_n} = 1.5 + (1/n) and {y_n} = 2.5 – (1/n) as converging sequences). But when I try to plug these values in f(x_n) and f(y_n), I get (2 – n) or an integral multiple of it in the denominator which doesn’t help me when taking the limit for either f(x_n) or f(y_n) when n→2.

I also tried some other combinations but I seem to run into the same problem, division by zero when taking the limit for either f(x_n) or f(y_n), can you please tell me if I am missing something or if I am doing something wrong here? Thanks in advance for all your help.